
theorem Th24: :: COROLLARY 4.13.
  for L be algebraic lower-bounded LATTICE ex g be Function of L,
  BoolePoset the carrier of CompactSublatt L st g is infs-preserving & g is
directed-sups-preserving & g is one-to-one & for x be Element of L holds g.x =
  compactbelow x
proof
  let L be algebraic lower-bounded LATTICE;
  deffunc F(Element of L) = compactbelow $1;
A1: for y be Element of L holds F(y) is Element of InclPoset Ids
  CompactSublatt L
  proof
    let y be Element of L;
    reconsider comy = compactbelow y as non empty directed Subset of L by
WAYBEL_8:def 4;
    reconsider comy as non empty Subset of CompactSublatt L by Th2;
    reconsider comy as non empty directed Subset of CompactSublatt L by
WAYBEL10:23;
    now
      let x1,z1 be Element of CompactSublatt L;
      reconsider x2 = x1, z2 = z1 as Element of L by YELLOW_0:58;
      assume x1 in comy & z1 <= x1;
      then x2 <= y & z2 <= x2 by WAYBEL_8:4,YELLOW_0:59;
      then
A2:   z2 <= y by ORDERS_2:3;
      z2 is compact by WAYBEL_8:def 1;
      hence z1 in comy by A2,WAYBEL_8:4;
    end;
    then comy is lower by WAYBEL_0:def 19;
    hence thesis by YELLOW_2:41;
  end;
  consider g1 be Function of L, InclPoset Ids CompactSublatt L such that
A3: for y be Element of L holds g1.y = F(y) from FUNCT_2:sch 9(A1);
  now
    let k be object;
    assume k in the carrier of InclPoset Ids CompactSublatt L;
    then k is Ideal of CompactSublatt L by YELLOW_2:41;
    then k is Element of BoolePoset the carrier of CompactSublatt L by
WAYBEL_8:26;
    hence k in the carrier of BoolePoset the carrier of CompactSublatt L;
  end;
  then the carrier of InclPoset Ids CompactSublatt L c= the carrier of
  BoolePoset the carrier of CompactSublatt L;
  then reconsider
  g = g1 as Function of L,BoolePoset the carrier of CompactSublatt
  L by FUNCT_2:7;
  take g;
A4: g1 is isomorphic by A3,Th16;
A5: InclPoset Ids CompactSublatt L is full infs-inheriting
directed-sups-inheriting SubRelStr of BoolePoset the carrier of CompactSublatt
  L by Th23;
  then ex g2 be Function of L,BoolePoset the carrier of CompactSublatt L st g1
  = g2 & g2 is infs-preserving by A4,Th21;
  hence g is infs-preserving;
  ex g3 be Function of L,BoolePoset the carrier of CompactSublatt L st g1
  = g3 & g3 is directed-sups-preserving by A4,A5,Th22;
  hence g is directed-sups-preserving;
  thus g is one-to-one by A4;
  let x be Element of L;
  thus thesis by A3;
end;
